# RH-W-11: Kernel Sensitivity and Regularity Duality v0.1

- 計畫：Riemann Hypothesis GAP Engineering
- 節點：`RH-W-11-KERNEL-SENSITIVITY-ORDER`
- 日期：2026-07-23
- 原始研究構想：Neo.K
- 數學工程、推導與實作：Aletheia (GPT-5.6 Thinking)
- 狀態：The activation order, regularity, and conservative tail bound order of the B-spline kernel family have been closed; does not constitute an RH proof

---

## 0. Problem of the Current Round

`RH-W-10` found that when using a cubic B-spline basis, its autocorrelation is a degree-$7$ B-spline; therefore, prime-$3$ appears after the support boundary only as

$$
\mu_+^7
$$

This makes the arithmetic threshold extremely smooth, but it also suppresses the local signal to approximately $10^{-28}$.

In this round, we no longer treat the cubic kernel as a fixed choice, but instead investigate the entire centered cardinal B-spline family:

1. How does the degree of the kernel determine the emergence order of the prime layer?
2. How does smoothness affect the Fourier decay and the Archimedean tail bound?
3. Does a single optimal kernel exist?
4. Can kernels of different degrees form a hierarchical sensing dictionary?

---

## 1. Centered Cardinal B-spline Family

The centered box function, with endpoint averaging, is

$$
\beta_0(x)=\mathbf 1_{[-1/2,1/2]}(x),
$$

and we recursively define

$$
\beta_m=\underbrace{\beta_0*\cdots*\beta_0}_{m+1\text{ times}}.
$$

Therefore:

$$
\operatorname{supp}(\beta_m)
=
\left[-\frac{m+1}{2},\frac{m+1}{2}\right],
$$

and $\beta_m$ is a degree-$m$ piecewise polynomial. For $m\ge1$,

$$
\beta_m\in C^{m-1}.
$$

Its truncated power representation is

$$
\beta_m(x)
=
\frac1{m!}
\sum_{k=0}^{m+1}
(-1)^k{m+1\choose k}
\left(x+\frac{m+1}{2}-k\right)_+^m.
$$

In the leftmost first segment, if $0<\varepsilon<1$, then

$$
\boxed{
\beta_m\left(-\frac{m+1}{2}+\varepsilon\right)
=
\frac{\varepsilon^m}{m!}
}.
$$

This first-segment formula is the origin of the prime boundary activation law.

---

## 2. Scaled and Translated Basis and Correlation Closure

Define the $L^2$-normalized basis

$$
v_{m,h,t}(x)
=
h^{-1/2}
\beta_m\left(\frac{x-t}{h}\right),
\qquad h>0.
$$

Let

$$
\widetilde v(x)=\overline{v(-x)}.
$$

From the convolution definition of B-splines, we immediately obtain:

$$
\boxed{
\beta_m*\beta_n=\beta_{m+n+1}
}.
$$

Thus, the cross-correlation at the same scale is

$$
\boxed{
(v_{m,h,t}*\widetilde v_{n,h,s})(x)
=
\beta_{m+n+1}
\left(
\frac{x-(t-s)}h
\right)
}.
$$

Let

$$
r=m+n+1.
$$

Then the cross-correlation has:

$$
\deg=r,
\qquad
C^{r-1}\text{ regularity},
\qquad
\text{support half-width }\frac{r+1}{2}h.
$$

Therefore, mixing different kernel degrees does not break closure; it merely maps the correlation kernel to another known B-spline degree.

---

## 3. General Prime-Power Soft Activation Law

Consider the negative logarithmic sample of a prime power $n=p^k$:

$$
-\log n.
$$

Assume it has just entered the left endpoint of the support of a cross-correlation, and let the dimensionless penetration depth be

$$
\varepsilon
=
\frac{\mu_+}{h},
\qquad
0<\varepsilon<1.
$$

where $\mu$ is the actual distance the sample exceeds the support boundary. By the first-segment formula, the corresponding discrete arithmetic matrix element is

$$
\boxed{
p_{n;m,n}(\varepsilon)
=
-\frac{\Lambda(n)}{\sqrt n}
\frac{\varepsilon^{m+n+1}}{(m+n+1)!}
}.
$$

Here $\Lambda(n)=\log p$.

Thus, the prime activation order of the cross-kernel is

$$
\boxed{r=m+n+1}.
$$

On the boundary, its regularity is

$$
\boxed{
C^{r-1}\text{ but not }C^r
}.
$$

### Autocorrelation Special Case

Taking $m=n$, we obtain

$$
\boxed{
r=2m+1}.
$$

Therefore:

| Basis degree $m$ | Autocorrelation degree | Prime activation order | Boundary regularity |
|---:|---:|---:|---:|
| 0 | 1 | 1 | $C^0$, not $C^1$ |
| 1 | 3 | 3 | $C^2$, not $C^3$ |
| 2 | 5 | 5 | $C^4$, not $C^5$ |
| 3 | 7 | 7 | $C^6$, not $C^7$ |
| 4 | 9 | 9 | $C^8$, not $C^9$ |
| 5 | 11 | 11 | $C^{10}$, not $C^{11}$ |

The cubic basis used in `RH-W-10` is exactly $m=3$, so its activation order is $7$.

---

## 4. Sensitivity-Regularity Duality

The Fourier transform of the B-spline under the centered convention satisfies

$$
\widehat{\beta_m}(\xi)
=
\left(
\frac{\sin(\xi/2)}{\xi/2}
\right)^{m+1}
$$

(differing by a phase or scale depending on the Fourier convention does not affect the decay order).

Thus, the Fourier decay of the $m,n$ cross-correlation is

$$
\boxed{
|\xi|^{-(m+n+2)}
=|\xi|^{-(r+1)}
}.
$$

On the other hand, the local amplitude at the prime boundary is

$$
\boxed{
O(\varepsilon^r)
}.
$$

The same integer $r$ simultaneously controls:

- The order to which the arithmetic threshold is flattened;
- The regularity of the correlation kernel;
- The Fourier decay;
- The conservative remainder order achievable by the term-by-term Laplace tail bounds in this engineering framework.

If the available derivatives are retained in the first local polynomial segment, the remainder of a single Laplace term can be controlled as

$$
O(a^{-(r+1)}),
$$

and for the tail sum with $a_k\asymp k$, we obtain the conservative order

$$
\boxed{
O(K^{-r})
}.
$$

Therefore, increasing the kernel degree simultaneously causes two opposing effects:

$$
\text{prime boundary signal becomes weaker},
$$

but

$$
\text{Archimedean/frequency-domain tail is easier to strictly control}.
$$

There is no free smoothness here.

---

## 5. Order-by-Order Compression Ratio

For the autocorrelation family, let

$$
a_m(\varepsilon)
=
\frac{\Lambda(n)}{\sqrt n}
\frac{\varepsilon^{2m+1}}{(2m+1)!}.
$$

Then the exact ratio between two adjacent orders is

$$
\boxed{
\frac{a_{m+1}}{a_m}
=
\frac{\varepsilon^2}{(2m+2)(2m+3)}
}.
$$

When $\varepsilon\ll1$, each increase in the basis degree not only suppresses $\varepsilon$ by two more powers but also divides by two growing integers.

Therefore, the sensitivity of high-degree kernels near the prime boundary drops extremely fast.

---

## 6. Strict Comparison with the Penetration Depth of `RH-W-10`

Adopting the post-boundary parameters from `RH-W-10`:

$$
h=\frac{87}{400},
\qquad
 d_+=\frac{73189}{320000}.
$$

The dimensionless penetration depth relative to the cubic boundary is

$$
\boxed{
\varepsilon
\approx
4.7510957190946486\times10^{-4}
}.
$$

Under the same $\varepsilon$, the magnitude of the strict interval midpoint for the prime-$3$ local element is:

| $m$ | Activation order | $|p_3|$ |
|---:|---:|---:|
| 0 | 1 | $3.0135444750\times10^{-4}$ |
| 1 | 3 | $1.1337411637\times10^{-11}$ |
| 2 | 5 | $1.2795918927\times10^{-19}$ |
| 3 | 7 | $6.8771698359\times10^{-28}$ |
| 4 | 9 | $2.1560797142\times10^{-36}$ |
| 5 | 11 | $4.4244540443\times10^{-45}$ |

Specifically:

$$
\boxed{
\frac{|p_3|_{m=1}}{|p_3|_{m=3}}
>
10^{16}
}.
$$

The actual calculation is approximately:

$$
1.64856\times10^{16}.
$$

Thus, the $10^{-28}$ observed in `RH-W-10` is not because prime-$3$ itself is inherently weak, but because the seventh-order soft activation of the cubic autocorrelation suppressed it by more than sixteen orders of magnitude.

---

## 7. Admissibility Boundary

The class $\mathcal W$ of admissible functions used by Bombieri/Clay has discontinuities of the first kind at finitely many points, and requires continuous differentiability and appropriate decay elsewhere. Compactly supported piecewise polynomial B-splines, after a multiplicative coordinate transformation, still belong to this generalized working class.

Therefore:

- $m=0$ can serve as a generalized $\mathcal W$ sensing kernel, but it does not belong to the $C_c^\infty$ core and must lock into the endpoint-averaging convention;
- $m=1$ is a continuous piecewise linear kernel, where the derivative only has finite jumps;
- $m\ge2$ provides progressively higher regularity;
- All these kernels still require the use of the full endpoint version of the Weil quadratic form, and cannot automatically assume double vanishing moments.

The high sensitivity of `m=0` does not mean it is optimal for certificate computation: its tail bound only exhibits low-order decay, which may require an extremely large cutoff or another closed-form treatment.

---

## 8. Conclusion of the Current Round

What this round closes is not RH positivity, but a kernel design problem:

$$
\boxed{
\text{The smoother the kernel, the more invisible the prime boundary;}
\quad
\text{the sharper the kernel, the more expensive the tail bound.}
}
$$

Therefore, no single degree dominates the other degrees across all engineering metrics.

The truly reasonable architecture is not to "find the optimal kernel," but to establish a multi-order dictionary:

- Low-degree kernels are responsible for sensing support events;
- High-degree kernels are responsible for stabilizing the background and high-precision certificates;
- Cross-correlations provide intermediate orders.

The next node is:

$$
\boxed{
\texttt{RH-W-12-MIXED-ORDER-DICTIONARY}
}.
$$

---

## 9. Scope Statement

This document proves the local scaling laws and engineering trade-offs within the centered cardinal B-spline family.

It does not:

- Find a Weil negative witness;
- Prove any infinite-dimensional positivity;
- Prove the RH;
- Interpret finite-dimensional near-zero modes as off-axis zeros of zeta.

---

## References

1. Enrico Bombieri, *The Riemann Hypothesis*, in *The Millennium Prize Problems*, Clay Mathematics Institute; which provides the test function class $\mathcal W$, the explicit formula, and the Weil criterion.
2. Masatoshi Suzuki, *Weil's quadratic form via the screw function*, arXiv:2606.09096, 2026; provides the fixed-interval and continuous function framework.
3. Akiva Groskin, *A finite Guinand–Weil dictionary and archimedean tail order for the truncated Weil quadratic form*, arXiv:2607.02828, 2026; emphasizes finite dictionaries, cutoff-free assembly, and calibratable tail bounds.